SCIENCE AWAKENING B.L.VAN DER WAERDEN SCIENCE AWAKENING I English translation by A rnold Dresden with additions of the author Fourth edition KLUWER ACADEMIC PUBLISHERS, DORDRECHT, THE NETHERLANDS SCHOLAR'S BOOKSHELF, PRINCETON JUNCTION, NEW JERSEY, U.S.A. Hardcover edition published throughout the world, exclusive of North America, by Kluwer Academic Publishers, Spuilboulevard 50, P.O. Box 17, 3300 AA Dordrecht, The Netherlands Paperback edition published throughout the world by The Scholar's Bookshelf, 51 Everett Drive, P.O. Box 179, Princeton Junction, New Jersey 08550, United States of America Hardcover edition published in North America by The Scholar's Bookshelf Copyright © 1975 by Noordhoff International Softcover reprint ofthe hardcover 4th edition 1975 Publishing, a division of Kluwer Academic Publishers, Dordrecht, The Netherlands All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form by any means, electronic, mechanical, photocopying, recording or otherwise without the prior permission of the copyright owner. First edition 1954 Second edition 1%1 Third edition 1969 Fourth edition 1975 Fifth edition 1988 Kluwer Academic Publishers ISBN-13: 978-94-010-7115-4 ~ISBN-13: 978-94-009-1379-0 DOl: 10.1007/978-94-009-1379-0 First Scholar's Bookshelf hardcover printing, 1988 First Scholar's Bookshelf paperback printing, 1988 PREFACE TO THE ENGLISH EDITION Soon after the publication of my"Ontwakende W etenschap"the need for an English translation was felt. We were very glad to find a translator fully familiar with the English and Dutch languages and with mathematical terminol· ogy. The publisher, Noordhoff, had the splendid idea to ask H. G. Beyen, professor of archeology, for his help in choosing a nice set of illustrations. It was a difficult task. The illustrations had to be both instructive and attractive, and they had t~ illustrate the history of science as well as the general background of ancient civilization. The publisher encouraged us to find better and still better illustrations, and he ordered photographs from all over the world, with never failing energy and enthusiasm. Mr. Beyen's highly instructive subscripts will help the reader to see the inter· relation between way of living, art, and science of the ancient world. Thanks are due to many correspondents, who have suggested additions and pointed out errors. Sections on Astrolabes and Stereographte Projection and on Archimedes' construction of the heptagon have been added. The sections on Perspective and on the Anaphorai of Hypsicles have been enlarged. In the second English edition I have incorporated an important discovery of P. Huber, which sheds new light upon the role of geometry In Babylonian algebra (see p. 73). The section on Heron's Metrics (see p. 277) was written anew, follOWing a suggestion of E. M. Bruins. Zurich. 1961 B. L. VAN DER WAERDEN PREFACE Why History of Mathematics? Every one knows that we are living in a technological era. But it is not often realized that our technology is based entirely on mathematics and physics. When we ride home on the streetcar in the evening, when we turn on the electric light and the radio, everything depends on cleverly constructed physical mechanisms based on mathematical calculations. But more than that! We owe to physics not only these pleasant articles of luxury, but, to a large extent, even our daily bread. Apart from the fact that our grains come to us, chiefly by steamer from overseas, our own agriculture would be far less productive without artificial fertilizers. Such fertilizers are chemical products, and chemistry depends on physics. l But science has not brought us blessings only. The destructive armaments which mankind uses at the present time to knock its own civilization to pieces are also products to which the development of mathematics and physics have inevitably led. Our spiritual life is also influenced hy science and technology, in a measure but rarely fully understood. The unprecedented growth of natural science in the 17th century was followed ineluctably by the rationalism of the 18th, by the deification of reason and the decline of religion; an analogous development had taken place earlier, in Greek times. In a similar manner, the triumphs of technology in the 19th century were followed in the 20th by the deification of technology. Unfor­ tunately, man seems to be overly inclined to deify whatever is powerful and successful. These considerations indicate that science has put its stamp on the whole of our life, material and spiritual, in its beneficent and in its evil aspects. Science is the most significant phenomenon of modern times, the principal ingredient of our civilization - alas! But if this be true, then the most important question for the history of culture is: How did our modern natural science come about? It will be conceded that most historical writings either do not consider this question at all, or else deal with it in a very unsatisfactory manner. For example, which are the histories of Greek culture that mention the names of Theaetetus and of Eudoxus, two of the greatest mathematicians of all times? Who realizes that, from the historical point of view, Newton is the most important figure of the 17th century? Every physicist will admit that the mechanics of Newton are the foundation of modern physics. Every astronomer knows that modern astronomy begins with Kepler and Newton. And every mathematician knows that the largest domain of 1 The ruder should bur in mind that this book was addressed originally to the public of the N,·thcrl.nds. 4 PREFACE modern mathematics, the part most important for physics, is Analysis, which has its roots in the Differential and Integral Calculus of Newton. Thus the work of Newton constitutes the foundation for by far the greater part of modern exact science. It was Newton who discovered the fundamental laws of motion, to which terrestrial as well as celestial objects are subject. He placed the crown on the task of renovating antique astronomy, begun by Copernicus and Kepler. He discovered a general method for solving all problems of differentiation and integration, whereas Archimedes, the greatest genius of antiquity, had not progressed beyond special methods for particular problems. The work of Newton can not be understood without a knowledge of antique science. Newton did not create in a void. Without the stupendous work of Ptole­ my, which completed and closed antique astronomy, Kepler's Astronomia Nova, and hence the mechanics of Newton, would have been impossible. Without the conic sectIOns of Apollonius, which Newton knew thoroughly, his deVelopment of the law of gravitation is equally unthinkable. And Newton's integral calculus can be understood only as a continuation of Archimedes' determination of areas and volumes. The history of mechanics as an exact science begins '.vith the laws of the lever, the laws of hydrostatics and the determinatIOn of mass centers by Archimedes. In short, all the developments which converge in the work of Newton,"1hose of mathematics, of mechanics and of astronomy, begin in Greece. The History of Greek Mathematics, from fhales to Apollonius, covers the four centuries from 600 B.C. to 200 B.C. Until recently, the first three of these four centuries were enveloped in twilight, because we possess only two original texts from this period: the fragment con­ cerning the lunules of Hippocrates and that of Archytas on the duplication of the cube. To this can be added two brief fragments of Archytas, a number of scattered communications of Plato, Aristotle, Pappus, Proclus and Eutocius, and a self­ contradictory set of Pythagorean legends. For this reason, the older works, such a~ Cantor's Geschichte der Mathematik, contain little more about this period than speculations concerning things of which we really do not know anything, such as, for example, the "Theorem of Pythagoras" In recent times however more light has penetrated into the darkness. In the first place, as a result of the indefatigable industry of Otto Neugebauer and his collaborators, we know now the mathematical cuneiform texts, which have thrown an entirely new light not only on the Theorem of Pythagoras, but especially on the earliest history of arithmetic and of algebra. Neugebauer, following in the tracks of Zeuthen, succeeded in discovering the hidden algebraic element in Greek mathematics and in demonstrating its connection with Babylonian algebra. No PREFACE 5 longer does the history of algebra begin with Diophantus; it starts 2000 years earlier in Mesopotamia. And, as to arithmetic, in 1937 Neugebauer wrote: "What is called Pythagorean in the Greek tradition, had probably better be called Babylonian"; and, a cuneiform text, concerning "Pythagorean numbers", dis­ covered in 1943, showed that he was entirely right. A second new impulse came from philosophically oriented philology. In 1927, Stenzel and Toeplitz, with Neugebauer, established the penodical "~ellen und Studien zur Geschichte der Mathematik, Astronomie und Physik". It was their purpose to get to know more about the philosophy of Plato by an analysis of the fundamental concepts of Greek mathematics, and, reciprocally, to learn more about Greek mathematics by means of an analysis of Plato. This method has enabled Becker, Reidemeister and others to obtain highly important results. At an earlier date, Eva Sachs had rescued the excellent mathematician Theaetetus from oblivion. Another very fertile method was the analysis of the Elements of Euclid. This work, written about 300 B.C., proves to be largely a compilation of mathematical fragments, quite diverse in calibre and quite varied in age. By carefully taking these fragments apart, by dusting them off and then replacing them in the mathe­ matical historical environment from which they had originally come, it has become possible to obtain a considerably clearer picture of Greek mathematICS of the years 500---300 B.C.